Generator form manipulation: Difference between revisions

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A [[canonical_form|canonical mapping form]] is an important standard to have as community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]]. Several such forms with different generator sizes have been presented, such as [[Normal_lists#Positive_generator_form|positive generator form]], [[Normal_lists#Equave-reduced_generator_form|equave-reduced generator form]], and [[Normal_lists#Minimal_generator_form|minimal-generator form]].

If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches.

For our purposes here we will be using the [[defactored Hermite form]] as the canonical form. If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches.

For example, the canonical form of [[Meantone_family#Meantone_.2812.2619.2C_2.3.5.29|5-limit meantone]] is {{ket|{{map|1 1 0}} {{map|0 1 4}}}}, and form has generators with sizes of approximately an [[octave]] and a [[perfect fifth]], respectively. But any pitch system constructed using an octave and a perfect fifth could also have been constructed using an octave and a [[perfect fourth]], because the perfect fourth is the [[octave complement]] of the perfect fifth. Specifically, any pitch we reached previously with a perfect fifth could be instead reached by going up an octave and down a perfect fourth. So in situations where we're approaching 5-limit meantone as a pitch system constructed by an octave and a perfect fourth, we might prefer to have the mapping in that form, which looks like {{ket|{{map|1 2 4}} {{map|0 -1 -4}}}}.

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 A [[canonical_form|canonical mapping form]] is an important standard to have as community for uniquely identifying [[temperaments]], but it is not the only mapping form one should ever need, because one may wish to use differently-sized [[generators]]. Several such forms with different generator sizes have been presented, such as [[Normal_lists#Positive_generator_form|positive generator form]], [[Normal_lists#Equave-reduced_generator_form|equave-reduced generator form]], and [[Normal_lists#Minimal_generator_form|minimal-generator form]].
-If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches.
+For our purposes here we will be using the [[defactored Hermite form]] as the canonical form. If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches.
 For example, the canonical form of [[Meantone_family#Meantone_.2812.2619.2C_2.3.5.29|5-limit meantone]] is {{ket|{{map|1 1 0}} {{map|0 1 4}}}}, and form has generators with sizes of approximately an [[octave]] and a [[perfect fifth]], respectively. But any pitch system constructed using an octave and a perfect fifth could also have been constructed using an octave and a [[perfect fourth]], because the perfect fourth is the [[octave complement]] of the perfect fifth. Specifically, any pitch we reached previously with a perfect fifth could be instead reached by going up an octave and down a perfect fourth. So in situations where we're approaching 5-limit meantone as a pitch system constructed by an octave and a perfect fourth, we might prefer to have the mapping in that form, which looks like {{ket|{{map|1 2 4}} {{map|0 -1 -4}}}}.